final version

RapportStage/Rapport.tex

@@ -140,6 +140,8 @@ Laboratoire de Chimie et Physique Quantiques
 
 \section*{Acknowledgments}
 
+I would like to sincerely thanks Pierre-François \textsc{LOOS} for supervising my internship. This first experience of academic research in theoretical chemistry confirmed my interest for this domain and taught me many things about theoretical chemistry but also about the job of researcher. I hope that I will have the chance to work in this group agian (and not in the context of a global pandemic if possible). I also thanks my three confined flatmates Tristan, Sylvio and especially Julie for bearing with me during those 3(+2) months.
+
 \tableofcontents
 
 \newpage
@@ -684,16 +686,14 @@ Exact 	& 9.783874 & 0.852781 & 0.391959 & 0.247898 & 0.139471 & 0.064525 & 0.005
 \label{tab:ERHFvsEUHF}
 \end{table}
 
-There is also another symmetry-broken solution for $R>75/38$ but this one corresponds to a maximum of the HF equations. This solution is associated with another type of symmetry breaking somewhat less known. It corresponds to a configuration where both electrons are on the same side of the sphere, in the same spatial orbital. This solution is called symmetry-broken RHF (sb-RHF). \antoine{The reasoning is counter-intuitive because the electrons tends to maximize their energy. If the orbitals are symmetric, the maximum is when the two electrons are in the p\textsubscript{z} orbital because it maximizes the kinetic energy. At the critical value of $R$, placing the two electrons in the same symmetry-broken orbital i.e., on the same side of the sphere gives a superior energy than the p\textsubscript{z}\textsuperscript{2} state. This is because it becomes more efficient to maximize the repulsion energy than the kinetic energy for $R>75/38$.}
-This configuration breaks the spatial symmetry of charge. Hence this symmetry breaking is associated with a charge density wave, the system oscillates between the situations with the two electrons on each side \cite{GiulianiBook}.
+There is also another symmetry-broken solution for $R>75/38$ but this one corresponds to a maximum of the HF equations. This solution is associated with another type of symmetry breaking somewhat less known. It corresponds to a configuration where both electrons are on the same side of the sphere, in the same spatial orbital. This solution is called symmetry-broken RHF (sb-RHF). The reasoning is counter-intuitive because the electrons tends to maximize their energy. The p\textsubscript{z} orbital is symmetric with respect to the center of the sphere. If the orbitals are symmetric, the maximum is when the two electrons are in the p\textsubscript{z} orbital because it maximizes the kinetic energy. At the critical value of $R$, placing the two electrons in the same symmetry-broken orbital i.e., on the same side of the sphere gives a superior energy than the p\textsubscript{z}\textsuperscript{2} state. Adding a s orbital on one side of the p\textsubscript{z} orbital to form a symmetry-broken orbital reduce the kinetic energy but increase the repulsion energy as the two electrons are more localized on one side of the sphere. It becomes more efficient to maximize the repulsion energy than the kinetic energy for $R>75/38$. This configuration breaks the spatial symmetry of charge. Hence this symmetry breaking is associated with a charge density wave, the system oscillates between the situations with the two electrons on each side \cite{GiulianiBook}.
 The energy associated with this sb-RHF solution reads
 \begin{equation}
 E_{\text{sb-RHF}}=\frac{75}{88R^3}+\frac{25}{22R^2}+\frac{91}{66R}.
 \end{equation}
-
 \begin{figure}[h!]
     \centering
-    \includegraphics[width=0.8\linewidth]{EsbHF.pdf}
+    \includegraphics[width=0.77\linewidth]{EsbHF.pdf}
     \caption{Energies of the five solutions of the HF equations (multiplied by $R^2$). The dotted curves correspond to the analytic continuation of the symmetry-broken solutions.}
     \label{fig:SpheriumNrj}
 \end{figure}
@@ -750,7 +750,6 @@ where $P_\ell$ are Legendre polynomials.
 Then, using this basis set we can compare the different partitioning of Sec.~\ref{sec:AlterPart}. Figure \ref{fig:RadiusPartitioning} shows the evolution of the radius of convergence $R_{\text{CV}}$ as a function of $R$ for the MP, the EN, the WC and the SC partitioning in a minimal basis (i.e., consisting of $P_0$ and $P_1$) of size $K = 2$, and in the same basis augmented with $P_2$ ($K = 3$). We see that, for the SC partitioning, $R_{\text{CV}}$ increases with $R$ whereas it is decreasing for the three others partitioning. This result is expected because the MP, EN, and WC partitioning use a weakly correlated reference so $\hH^{(0)}$ is a good approximation for small $R$. On the contrary, the SC partitioning consider naturally a strongly correlated reference so the SC series converges far better when the electron are strongly correlated, i.e., when $R$ is large in the spherium model.
 
 Interestingly, the radius of convergence associated with the SC partitioning is greater than one for a greater range of radii for $K = 2$ than $K = 3$. 
-In the complete basis limit, the radius of convergence of the SC partitioning is greater than unity only for very large value of $R$.
 
 \begin{figure}
     \centering
@@ -764,6 +763,7 @@ The MP partitioning is always better than WC in Fig.~\ref{fig:RadiusPartitioning
 Interestingly, it can be proved that the $m$th order energy of the WC series can be obtained as a Taylor series of MP$m$ with respect to $R$. 
 It seems that the EN is better than MP for very small $R$ in the minimal basis. In fact, it is just an artefact of the minimal basis because, for $K = 3$, the MP series has a greater radius of convergence for all values of $R$. It holds true for $K>3$.
 
+
 Figure \ref{fig:RadiusBasis} shows that the radius of convergence is not very sensitive to the size of the basis set. The CSFs have all the same spin and spatial symmetries so we expect that the singularities obtained within this basis set will be $\alpha$ singularities. Table \ref{tab:SingAlpha} shows that the singularities considered in this case are indeed $\alpha$ singularities. This is consistent with the observation of Goodson and Sergeev \cite{Goodson_2004} who stated that $\alpha$ singularities are relatively insensitive to the basis set size. The discontinuities observed in Fig.~\ref{fig:RadiusBasis} for the MP partitioning are due to changes in dominant singularity. We can observe this change in Table \ref{tab:SingAlpha}, the value for $R=1$ and $R=2$ are respectively in the positive and negative planes.
 
 \begin{figure}[h!]